Change UV Positions for Linked Objects?
Use Different UV Maps

It is possible. The list of UV maps is attached to Mesh data, each objects sharing that same data can't have a different set of UV maps. But each object's material can be attached to itself instead of its data, and its texture can use any available UV map.

First, create all UV maps to be used in the same object data (max. 8). 



Then create different materials for each objects, set the material's link property to Object. 



Then create a new image texture for each materials, referring to the same texture file but using different UV map. Its limitation is the amount of UV maps can be created for a single object data, 8 at maximum.

Use Different Textures

Otherwise, why not create different texture files using the same UV map? I mean, I imagine you wouldn't vary seam placement, just shifting the whole UV coordinates around. Wouldn't that be too much work and inflexible (e.g. modifying every single map when the mesh object is edited)?

Using one UV map as a template is easier, IMHO. And you're free to create as many texture variations as you need.